The doctrine of permutations and combinations, being an essential and fundamental part of the doctrine of chances; as it is delivered by Mr. James Bernoulli, in his excellent treatise on the doctrine of chances, intitled, Ars conjectandi, and by the celebrated Dr. John Wallis, of Oxford, in a tract intitled from the subject, and published at the end of his Treatise on algebra: in the former of which tracts is contained, a demonstration of Sir Isaac Newton's famous binomial theorem, in the cases of integral powers, and of the reciprocals of integral powers. Together with some other useful mathematical tracts / [by various authors] Published by Francis Maseres, esq.

Date:
1795
Catalogue details

Licence: Public Domain Mark

Credit: The doctrine of permutations and combinations, being an essential and fundamental part of the doctrine of chances; as it is delivered by Mr. James Bernoulli, in his excellent treatise on the doctrine of chances, intitled, Ars conjectandi, and by the celebrated Dr. John Wallis, of Oxford, in a tract intitled from the subject, and published at the end of his Treatise on algebra: in the former of which tracts is contained, a demonstration of Sir Isaac Newton's famous binomial theorem, in the cases of integral powers, and of the reciprocals of integral powers. Together with some other useful mathematical tracts / [by various authors] Published by Francis Maseres, esq. Source: Wellcome Collection.

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    and the fecond term of the fifth vertical cokimn is 5, and the fecond term of the fixth vertical column is 6 ; and the fame thing may be obferved in all the following columns. This property is too evident to need a proof. C4. The fourth property. Every term in the foregoing table is equal to the fum of all the terms that ftand above it in the next preceeding vertical column. Thus, for example, 56 (which is the fixth fignificant term in the fourth vertical column) is equal to the fum of i, 3, 6, 10, 15, and 21, which are the firfl; fix fignificant terms of the next preceed- ing, or third, vertical column, and which all ftand above the faid term 56, or above the term 28 in the third vertical column, which ftands even with the faid term 56 in the fourth column. This property is manifeft from art. 40, 41, 42, &c. — — 4^* 155. The fifth property. Every term in the table, after the firft term 1 in each horizontal row of terms, is equal to the fum of the two terms that ftand immediately above it in the fame vertical column and in the next preceeding vertical column. Thus, for example, 56 (which is the 9th term in the fourth vertical column, including the cyphers, or the fixth term exclufive of the cyphers) is equal to the funi of 35, which is the term next above 56 in the fame vertical column, and 21, which is the term next above 56 in the next preceeding or third vertical column. And, from the manner in which the table is formed, the fame thing is dent of every other term in the table. 56. The fixth property. The terms of every tranfverfe, or horizontal, column increafe gradually from i to a certain magnitude, and then decreafe again by the fame degrees to 1, fo as to make the terms that are equidiftant from the beginning and the end of the column be equal to each other. Thus, for example, the terms of the yth tranfverfe, or ho- rizontal columnare i, 6, 15, 20, 15, 6, i, in which the firft and the laft term arc, both of them, i, the fecond ' term